Place a rack in the center of the oven and preheat the oven to 350°F Line 24 cupcake cups with paper liners. Set the pans aside.
Prepare the cupcake batter:
With a citrus zester or fine grater, zest the lime and reserve one teaspoon. Cut the lime in half and squeeze the juice over a strainer into a small bowl; you should have two tablespoons. Set aside.
Separate three of the eggs, place the whites in a large clean mixing bowl and the yolks in another mixing bowl.
Add the remaining egg, cake mix, pudding mix, water, oil, coconut flavoring and lime zest and juice to the yolks. Blend with an electric mixer on low speed for 30 seconds. Stop the machine and scrape down the sides of the bowl with a rubber spatula. Increase the mixer speed to medium and beat 2 minutes more, scraping down the sides again if needed. The batter should look well blended.
Spoon or scoop 1/4 cup batter into each lined cupcake cup, filling it two thirds of the way full. Place the pans in the oven.
Bake the cupcakes until they are lightly golden and spring back when lightly pressed with your finger, 18 to 22 minutes. Remove the pans from the oven and place them on wire racks to cool for five minutes. Increase the oven heat to 450°F.
Run a dinner knife around the edges of the cupcake liners, lift the cupcakes up from the bottoms of the cups using the end of the knife, and pick them out of the cups carefully with your fingertips. Place them on a wire rack to cool for 15 minutes before filling.
Meanwhile, prepare the filling: Place the sweetened condensed milk and Key lime juice in a small bowl and stir to combine. Spoon the mixture into a pastry bag fitted with a medium tip. Press the tip 1/4 inch into the top center of each cooked cupcake and squeeze the bag to release about 1 tablespoon filling into the cupcake. Scrape off any excess filling with a rubber spatula. Place the cooled cupcakes nearly side by side on a baking sheet.
Prepare the meringue:
Add the cream of tartar to the reserved egg whites and beat with an electric mixer on high speed until frothy, 45 seconds.
Add the coconut flavoring and one tablespoon of the sugar. Continue beating on high, adding one tablespoon sugar at a time, until the whites form stiff peaks (if you pull the beaters from the meringue, the meringue should extend from the beaters in straight stiff peaks).
Spoon two tablespoons meringue onto the top of each cupcake and swirl to spread it just to the edges with a short metal spatula or a spoon. Place the pan in the oven. Bake until the meringue is light nutty brown in color, five to six minutes.
Run a dinner knife around the edges of the cupcake liners, lift the cupcakes up from the bottoms of the cups using the end of the knife, and pick them out of the cups carefully with your fingers. Serve at once, or allow the cupcakes to cool 10 minutes, then serve.